= Intrinsic phase averages in a two-fluid pipe model
{title2=$\alpha=\langle\chi_1\rangle,\quad\rho_1=\langle\rho\chi_1\rangle/\alpha,\quad u_1=\langle\rho v_x\chi_1\rangle/(\alpha\rho_1)$}
Here $\langle\cdot\rangle$ means integration over a fixed pipe cross-section divided by its area, and $\chi_1$ is the indicator of one immiscible fluid. The <volume fraction> is $\alpha$; intrinsic <density> averages over the phase-occupied area rather than over the whole section. The <velocity> must be <density>-weighted for its product with mean phase <density> to represent <mass flux> exactly.
If there is no interphase mass transfer, integrating $\partial_t(\rho\chi_1)+\nabla\cdot(\rho v\chi_1)=0$ over a section with impermeable outer wall gives $\partial_t(\rho_1\alpha)+\partial_x(\rho_1\alpha u_1)=0$. The second phase uses $1-\alpha$ and its own intrinsic averages. For constant phase <density> the <velocity> reduces to the ordinary phase-area average. These definitions apply where the corresponding phase fraction is positive; zero-fraction intrinsic quantities need a separate convention.
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